Dynamics of a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation

1990 ◽  
Vol 57 (1) ◽  
pp. 209-217 ◽  
Author(s):  
Kazuyuki Yagasaki ◽  
Masaru Sakata ◽  
Koji Kimura
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Invariant Tori ◽  
Homoclinic Orbits ◽  
External Excitation ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Average System ◽  
Theoretical Results

In this paper we study the dynamics of a weakly nonlinear single-degree-of-freedom system subjected to combined parametric and external excitation. The averaging method is used to establish the existence of invariant tori and analyze their stability. Furthermore, by applying the Melnikov technique to the average system it is shown that there exist transverse homoclinic orbits resulting in chaotic dynamics. Numerical simulation results are also given to demonstrate the theoretical results.

Chaos in a Weakly Nonlinear Oscillator With Parametric and External Resonances

1991 ◽  
Vol 58 (1) ◽  
pp. 244-250 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Invariant Torus ◽  
Chaotic Dynamics ◽  
Averaging Method ◽  
Double Resonance ◽  
External Excitation ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
External Forces

This paper describes a study of the chaotic dynamics of a weakly nonlinear single degree-of-freedom system subjected to combined parametric and external excitation. We consider a case of double resonance in which primary resonances, with respect to parametric and external forces, exist simultaneously. By using the averaging method and Melnikov’s technique, it is shown that chaos may occur in certain parameter regions. These chaotic motions result from the existence of orbits homoclinic to a normally hyperbolic invariant torus which corresponds to a hyperbolic periodic orbit in the averaged system. The mechanism and structure of chaos in this situation are also described. Furthermore, the existence of steady-state chaos is demonstrated by numerical simulation.

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750205 ◽  
Author(s):  
Tonghua Zhang ◽  
Jibin Li
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Torus Knot ◽  
Cubic System ◽  
Theoretical Results

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

Application of the Random Melnikov Method for Single-Degree-of-Freedom Vessel Rolls Motion

2010 ◽  
Author(s):  
Kaiye Hu ◽  
Yong Ding ◽  
Hongwei Wang ◽  
Jide Li
Keyword(s):  
Global Stability ◽  
Global Bifurcation ◽  
Melnikov Method ◽  
Threshold Value ◽  
Homoclinic Orbits ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Bounded Noise ◽  
Single Degree Of Freedom ◽  
Single Degree

Basing on the nonlinear dynamics theory, the global stability of ship in stochastic beam sea is researched by the global bifurcation method. In this paper, bounded noise is first briefly introduced. Bounded noise is a harmonic function with constant random frequency and phase. It has finite power and its spectral shape can be made to fit a target spectrum, such as Pierson-Moskowitz spectrum, by adjusting its parameters. This paper considered the stochastic excitation term as bounded noise and the influence of nonlinear damping and nonlinear righting moment, setup the random single degree of freedom nonlinear rolling equation. Then the random Melnikov process for the nonlinear system with homoclinic orbits under both dissipative and bounded noise perturbations is derived. The random Melnikov mean-square criterion is used to analysis the global stability of this system. The research indicates that the bounded noise can approximately simulate the wave excitation and if the noise exceeds the threshold value, the ship will undergo stochastic chaotic motion. That will lead ships to instability and even to capsizing.

RESONANCE CONTROL FOR A FORCED SINGLE-DEGREE-OF-FREEDOM NONLINEAR SYSTEM

2004 ◽  
Vol 14 (04) ◽  
pp. 1423-1429 ◽  
Author(s):  
ANDREW Y. T. LEUNG ◽  
JIN CHEN JI ◽  
GUANRONG CHEN
Keyword(s):  
Feedback Control ◽  
Nonlinear System ◽  
Singularity Theory ◽  
Degree Of Freedom ◽  
Theory Approach ◽  
Nonlinear Feedback ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Resonance Control

The main characteristic of a forced single-degree-of-freedom weakly nonlinear system is determined by its primary, super- and sub-harmonic resonances. A nonlinear parametric feedback control is proposed to modify the steady-state resonance responses, thus to reduce the amplitude of the response and to eliminate the saddle-node bifurcations that take place in the resonance responses. The nonlinear gain of the feedback control is determined by analyzing the bifurcation diagrams associated with the corresponding frequency-response equation, from the singularity theory approach. It is shown by illustrative examples that the proposed nonlinear feedback is effective for controlling three kinds of resonance responses.

scholarly journals On the critical forcing amplitude of forced nonlinear oscillators

2013 ◽  
Vol 3 (4) ◽  
Author(s):  
Mariano Febbo ◽  
Jinchen Ji
Keyword(s):  
Primary Resonance ◽  
Nonlinear Oscillators ◽  
Analytical Form ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Saddle Node ◽  
Basis Method

AbstractThe steady-state response of forced single degree-of-freedom weakly nonlinear oscillators under primary resonance conditions can exhibit saddle-node bifurcations, jump and hysteresis phenomena, if the amplitude of the excitation exceeds a certain value. This critical value of excitation amplitude or critical forcing amplitude plays an important role in determining the occurrence of saddle-node bifurcations in the frequency-response curve. This work develops an alternative method to determine the critical forcing amplitude for single degree-of-freedom nonlinear oscillators. Based on Lagrange multipliers approach, the proposed method considers the calculation of the critical forcing amplitude as an optimization problem with constraints that are imposed by the existence of locations of vertical tangency. In comparison with the Gröbner basis method, the proposed approach is more straightforward and thus easy to apply for finding the critical forcing amplitude both analytically and numerically. Three examples are given to confirm the validity of the theoretical predictions. The first two present the analytical form for the critical forcing amplitude and the third one is an example of a numerically computed solution.

Chaotic Dynamics of a Quasi-Periodically Forced Beam

1992 ◽  
Vol 59 (1) ◽  
pp. 161-167 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Single Mode ◽  
Homoclinic Orbits ◽  
Equation Of Motion ◽  
Degree Of Freedom ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Higher Modes ◽  
Straight Beam

A straight beam with fixed ends, forced with two frequencies is considered. By using Galerkin’s method, the equation of motion of the beam is reduced to a finite degree-of-freedom system. The resulting equation is transformed into a multi-frequency system and the averaging method is applied. It is shown, by using Melnikov’s method, that there exist transverse homoclinic orbits in the averaged system associated with the first-mode equation. This implies that chaotic motions may occur in the single-mode equation. Furthermore, the effect of higher modes and the implications of this result for the full beam motions are described.

scholarly journals A Clamped Bar Model for the Sompoton Vibrator

2013 ◽  
Vol 38 (3) ◽  
pp. 425-432
Author(s):  
Tee Hao Wong ◽  
Jedol Dayou ◽  
M.C.D. Ngu ◽  
Jackson H.W. Chang ◽  
Willey Y.H. Liew
Keyword(s):  
Fundamental Frequency ◽  
Experimental Measurement ◽  
Musical Instruments ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Exact Analysis ◽  
Air Jet ◽  
Production Errors ◽  
Theoretical Results ◽  
Fundamental Resonance Frequency

Abstract The sompoton is one of famous traditional musical instruments in Sabah. This instrument consists of several parts with the vibrator being the most important one. In this paper, the vibrator is modeled as a clamped bar with a uniformly distributed mass. By means of this model, the fundamental frequency is analyzed with the use of an equivalent single degree of freedom system (SDOF) and exact analysis. The vibrator is made of aluminum in different sizes and is excited using a constant air jet to obtain its fundamental resonance frequency. The fundamental frequency obtained from the experimental measurement is compared with the theoretical values calculated based on the equivalent SDOF and exact analysis theories. It is found that the exact analysis gives a closer value to the experimental results as compared to the SDOF system. Although both the experimental and theoretical results exhibit the same trend, they are different in magnitude. To overcome the differences in both theories, a correction factor is added to account for the production errors.

Ideal Elastic-Plastic Oscillators Subjected to Stochastic Input

1999 ◽  
Author(s):  
S. Caddemi ◽  
M. Di Paola
Keyword(s):  
External Noise ◽  
Spectral Density Function ◽  
Power Spectral Density Function ◽  
External Excitation ◽  
Homogeneous Poisson Process ◽  
Single Degree Of Freedom ◽  
Elastic Plastic ◽  
Single Degree ◽  
Power Spectral ◽  
Probabilistic Response

Abstract The paper deals with the evaluation of the probabilistic response of an ideal elastic-plastic single degree of freedom oscillator subjected to a normal white noise. The analysis has been conducted on the hypothesis that accumulated plastic displacements are a compound homogeneous Poisson process independent of the external excitation. In this case plastic displacements can be treated as an additional external noise, to be identified, acting on a linear system. In the paper a time domain approach to obtain the two variable non stationary correlation function is proposed. Hence the evolutionary power spectral density function is also obtained. A numerical example is presented in order to show the accuracy of the presented procedure in comparison to Monte Carlo simulations.

Galloping of a Wind-Excited Tower Under Internal Parametric Damping

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Lahcen Mokni ◽  
Ilham Kirrou ◽  
Mohamed Belhaq
Keyword(s):  
Periodic Solutions ◽  
Perturbation Analysis ◽  
Degree Of Freedom ◽  
External Excitation ◽  
Single Degree Of Freedom ◽  
Turbulent Wind ◽  
Single Degree

The effect of harmonic internal parametric damping (IPD) on the amplitude and the onset of the periodic galloping of a tower is investigated in the presence of steady and unsteady wind. The structure is modeled by a lumped single degree of freedom (sdof) equation and attention is focused on the cases where the unsteady (turbulent) wind activates the external excitation, the parametric one, or both. A perturbation analysis is performed to approximate periodic solutions and the effect of the IPD on the amplitude and the onset of periodic galloping is examined in different cases of loading. It is shown that the IPD substantially improves the reduction in the galloping amplitude for all cases of loading and it has no influence on the galloping onset.

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